BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 283 



resulting points in turn, so long as they are singular. If after a finite 

 number of such processes, all the resulting points are regular, then by 

 combining the results it is assumed that the neighborhood of the origi- 

 nal point is represented by the domains of a finite number of regular 

 points, and so by a finite number of parametric formulae as desired. 



3. Proof that a Finite Number of the Processes of 1 will be Sufficient 

 to make all Points in 2 Regular. Starting with the surface 



f(u, v, w)=0, (f ) 



in which the singular point considered is at the origin, the transfor- 

 mations in 1, 1) and 2) are combined in the form 



u = (out + fro- + yOn 



v = (a 2 T + /? 2 o-+ 72 )£>- (g) 



W= (a 3 r + fi 3 o- + y 3 )0 



We can assume that 



y-2 + , y 8 4= ° 



by making, if necessary, upon / (it, r, w) a suitable homogeneous 

 linear transformation. Then the next set of transformations, in 2, can 

 be expressed in the form 



r=(a 1 'r 1 + /Vo-! + y/Ki ) 



<r= (a 2 ' Tl + A/o-! + y 2 ') & V (h) 



C=(o,'t 1 + /V^ + y/Kx J 



in which y 3 ' -^ 0,* and the later sets of transformations are of the same 

 type with the corresponding y 3 's : 



y 3 "^0, y 3 "'t0, etc. 



So we consider a succession of transformations of type (g), which give 

 a succession of surfaces with multiple points each of order m. These 

 transformations will combine in the form 



«=[yiy 3 'y 3 " ys (r) + (*„ °v, £■)]£• = [A + (r r , <r r ,£ r )] {,A 



V = [y 2 y 3 'y 3 " ya" 1 + (r„ <r,, £) ] t r = [T\ + (r„ cr,, £.)] l r \ (i) 



«> = [y 3 y 3 ' y 3 " y 3 M + (r,, <r r , £.)] C = [r, + (r„ <r r , £.)] C ) 



where the symbol (t,., <r r , £ r ) represents in the expression in which it 

 occurs all of the variable terms, and r 2 =}= 0, T 3 4= 0. 



* To secure this, Kobb makes unwarranted use of a quadratic transformation, 

 which, however, might be replaced by a homogeneous linear transformation. He 

 also overlooks one class of transformations which will arise (see 4). 



